<!DOCTYPE html>
<html lang="en">
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<meta charset="UTF-8">
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<title>mathFunctions(3)</title>
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/*! normalize.css v2.1.2 | MIT License | git.io/normalize */
/* ========================================================================== HTML5 display definitions ========================================================================== */
/** Correct `block` display not defined in IE 8/9. */
article, aside, details, figcaption, figure, footer, header, hgroup, main, nav, section, summary { display: block; }

/** Correct `inline-block` display not defined in IE 8/9. */
audio, canvas, video { display: inline-block; }

/** Prevent modern browsers from displaying `audio` without controls. Remove excess height in iOS 5 devices. */
audio:not([controls]) { display: none; height: 0; }

/** Address `[hidden]` styling not present in IE 8/9. Hide the `template` element in IE, Safari, and Firefox < 22. */
[hidden], template { display: none; }

script { display: none !important; }

/* ========================================================================== Base ========================================================================== */
/** 1. Set default font family to sans-serif. 2. Prevent iOS text size adjust after orientation change, without disabling user zoom. */
html { font-family: sans-serif; /* 1 */ -ms-text-size-adjust: 100%; /* 2 */ -webkit-text-size-adjust: 100%; /* 2 */ }

/** Remove default margin. */
body { margin: 0; }

/* ========================================================================== Links ========================================================================== */
/** Remove the gray background color from active links in IE 10. */
a { background: transparent; }

/** Address `outline` inconsistency between Chrome and other browsers. */
a:focus { outline: thin dotted; }

/** Improve readability when focused and also mouse hovered in all browsers. */
a:active, a:hover { outline: 0; }

/* ========================================================================== Typography ========================================================================== */
/** Address variable `h1` font-size and margin within `section` and `article` contexts in Firefox 4+, Safari 5, and Chrome. */
h1 { font-size: 2em; margin: 0.67em 0; }

/** Address styling not present in IE 8/9, Safari 5, and Chrome. */
abbr[title] { border-bottom: 1px dotted; }

/** Address style set to `bolder` in Firefox 4+, Safari 5, and Chrome. */
b, strong { font-weight: bold; }

/** Address styling not present in Safari 5 and Chrome. */
dfn { font-style: italic; }

/** Address differences between Firefox and other browsers. */
hr { -moz-box-sizing: content-box; box-sizing: content-box; height: 0; }

/** Address styling not present in IE 8/9. */
mark { background: #ff0; color: #000; }

/** Correct font family set oddly in Safari 5 and Chrome. */
code, kbd, pre, samp { font-family: monospace, serif; font-size: 1em; }

/** Improve readability of pre-formatted text in all browsers. */
pre { white-space: pre-wrap; }

/** Set consistent quote types. */
q { quotes: "\201C" "\201D" "\2018" "\2019"; }

/** Address inconsistent and variable font size in all browsers. */
small { font-size: 80%; }

/** Prevent `sub` and `sup` affecting `line-height` in all browsers. */
sub, sup { font-size: 75%; line-height: 0; position: relative; vertical-align: baseline; }

sup { top: -0.5em; }

sub { bottom: -0.25em; }

/* ========================================================================== Embedded content ========================================================================== */
/** Remove border when inside `a` element in IE 8/9. */
img { border: 0; }

/** Correct overflow displayed oddly in IE 9. */
svg:not(:root) { overflow: hidden; }

/* ========================================================================== Figures ========================================================================== */
/** Address margin not present in IE 8/9 and Safari 5. */
figure { margin: 0; }

/* ========================================================================== Forms ========================================================================== */
/** Define consistent border, margin, and padding. */
fieldset { border: 1px solid #c0c0c0; margin: 0 2px; padding: 0.35em 0.625em 0.75em; }

/** 1. Correct `color` not being inherited in IE 8/9. 2. Remove padding so people aren't caught out if they zero out fieldsets. */
legend { border: 0; /* 1 */ padding: 0; /* 2 */ }

/** 1. Correct font family not being inherited in all browsers. 2. Correct font size not being inherited in all browsers. 3. Address margins set differently in Firefox 4+, Safari 5, and Chrome. */
button, input, select, textarea { font-family: inherit; /* 1 */ font-size: 100%; /* 2 */ margin: 0; /* 3 */ }

/** Address Firefox 4+ setting `line-height` on `input` using `!important` in the UA stylesheet. */
button, input { line-height: normal; }

/** Address inconsistent `text-transform` inheritance for `button` and `select`. All other form control elements do not inherit `text-transform` values. Correct `button` style inheritance in Chrome, Safari 5+, and IE 8+. Correct `select` style inheritance in Firefox 4+ and Opera. */
button, select { text-transform: none; }

/** 1. Avoid the WebKit bug in Android 4.0.* where (2) destroys native `audio` and `video` controls. 2. Correct inability to style clickable `input` types in iOS. 3. Improve usability and consistency of cursor style between image-type `input` and others. */
button, html input[type="button"], input[type="reset"], input[type="submit"] { -webkit-appearance: button; /* 2 */ cursor: pointer; /* 3 */ }

/** Re-set default cursor for disabled elements. */
button[disabled], html input[disabled] { cursor: default; }

/** 1. Address box sizing set to `content-box` in IE 8/9. 2. Remove excess padding in IE 8/9. */
input[type="checkbox"], input[type="radio"] { box-sizing: border-box; /* 1 */ padding: 0; /* 2 */ }

/** 1. Address `appearance` set to `searchfield` in Safari 5 and Chrome. 2. Address `box-sizing` set to `border-box` in Safari 5 and Chrome (include `-moz` to future-proof). */
input[type="search"] { -webkit-appearance: textfield; /* 1 */ -moz-box-sizing: content-box; -webkit-box-sizing: content-box; /* 2 */ box-sizing: content-box; }

/** Remove inner padding and search cancel button in Safari 5 and Chrome on OS X. */
input[type="search"]::-webkit-search-cancel-button, input[type="search"]::-webkit-search-decoration { -webkit-appearance: none; }

/** Remove inner padding and border in Firefox 4+. */
button::-moz-focus-inner, input::-moz-focus-inner { border: 0; padding: 0; }

/** 1. Remove default vertical scrollbar in IE 8/9. 2. Improve readability and alignment in all browsers. */
textarea { overflow: auto; /* 1 */ vertical-align: top; /* 2 */ }

/* ========================================================================== Tables ========================================================================== */
/** Remove most spacing between table cells. */
table { border-collapse: collapse; border-spacing: 0; }

meta.foundation-mq-small { font-family: "only screen and (min-width: 768px)"; width: 768px; }

meta.foundation-mq-medium { font-family: "only screen and (min-width:1280px)"; width: 1280px; }

meta.foundation-mq-large { font-family: "only screen and (min-width:1440px)"; width: 1440px; }

*, *:before, *:after { -moz-box-sizing: border-box; -webkit-box-sizing: border-box; box-sizing: border-box; }

html, body { font-size: 100%; }

body { background: white; color: #222222; padding: 0; margin: 0; font-family: "Helvetica Neue", "Helvetica", Helvetica, Arial, sans-serif; font-weight: normal; font-style: normal; line-height: 1; position: relative; cursor: auto; }

a:hover { cursor: pointer; }

img, object, embed { max-width: 100%; height: auto; }

object, embed { height: 100%; }

img { -ms-interpolation-mode: bicubic; }

#map_canvas img, #map_canvas embed, #map_canvas object, .map_canvas img, .map_canvas embed, .map_canvas object { max-width: none !important; }

.left { float: left !important; }

.right { float: right !important; }

.text-left { text-align: left !important; }

.text-right { text-align: right !important; }

.text-center { text-align: center !important; }

.text-justify { text-align: justify !important; }

.hide { display: none; }

.antialiased { -webkit-font-smoothing: antialiased; }

img { display: inline-block; vertical-align: middle; }

textarea { height: auto; min-height: 50px; }

select { width: 100%; }

object, svg { display: inline-block; vertical-align: middle; }

.center { margin-left: auto; margin-right: auto; }

.spread { width: 100%; }

p.lead, .paragraph.lead > p, #preamble > .sectionbody > .paragraph:first-of-type p { font-size: 1.21875em; line-height: 1.6; }

.subheader, .admonitionblock td.content > .title, .audioblock > .title, .exampleblock > .title, .imageblock > .title, .listingblock > .title, .literalblock > .title, .stemblock > .title, .openblock > .title, .paragraph > .title, .quoteblock > .title, table.tableblock > .title, .verseblock > .title, .videoblock > .title, .dlist > .title, .olist > .title, .ulist > .title, .qlist > .title, .hdlist > .title { line-height: 1.4; color: black; font-weight: 300; margin-top: 0.2em; margin-bottom: 0.5em; }

/* Typography resets */
div, dl, dt, dd, ul, ol, li, h1, h2, h3, #toctitle, .sidebarblock > .content > .title, h4, h5, h6, pre, form, p, blockquote, th, td { margin: 0; padding: 0; direction: ltr; }

/* Default Link Styles */
a { color: #0068b0; text-decoration: none; line-height: inherit; }
a:hover, a:focus { color: #333333; }
a img { border: none; }

/* Default paragraph styles */
p { font-family: Noto, sans-serif; font-weight: normal; font-size: 1em; line-height: 1.6; margin-bottom: 0.75em; text-rendering: optimizeLegibility; }
p aside { font-size: 0.875em; line-height: 1.35; font-style: italic; }

/* Default header styles */
h1, h2, h3, #toctitle, .sidebarblock > .content > .title, h4, h5, h6 { font-family: Noto, sans-serif; font-weight: normal; font-style: normal; color: black; text-rendering: optimizeLegibility; margin-top: 0.5em; margin-bottom: 0.5em; line-height: 1.2125em; }
h1 small, h2 small, h3 small, #toctitle small, .sidebarblock > .content > .title small, h4 small, h5 small, h6 small { font-size: 60%; color: #4d4d4d; line-height: 0; }

h1 { font-size: 2.125em; }

h2 { font-size: 1.6875em; }

h3, #toctitle, .sidebarblock > .content > .title { font-size: 1.375em; }

h4 { font-size: 1.125em; }

h5 { font-size: 1.125em; }

h6 { font-size: 1em; }

hr { border: solid #dddddd; border-width: 1px 0 0; clear: both; margin: 1.25em 0 1.1875em; height: 0; }

/* Helpful Typography Defaults */
em, i { font-style: italic; line-height: inherit; }

strong, b { font-weight: bold; line-height: inherit; }

small { font-size: 60%; line-height: inherit; }

code { font-family: Consolas, "Liberation Mono", Courier, monospace; font-weight: normal; color: #264357; }

/* Lists */
ul, ol, dl { font-size: 1em; line-height: 1.6; margin-bottom: 0.75em; list-style-position: outside; font-family: Noto, sans-serif; }

ul, ol { margin-left: 1.5em; }
ul.no-bullet, ol.no-bullet { margin-left: 1.5em; }

/* Unordered Lists */
ul li ul, ul li ol { margin-left: 1.25em; margin-bottom: 0; font-size: 1em; /* Override nested font-size change */ }
ul.square li ul, ul.circle li ul, ul.disc li ul { list-style: inherit; }
ul.square { list-style-type: square; }
ul.circle { list-style-type: circle; }
ul.disc { list-style-type: disc; }
ul.no-bullet { list-style: none; }

/* Ordered Lists */
ol li ul, ol li ol { margin-left: 1.25em; margin-bottom: 0; }

/* Definition Lists */
dl dt { margin-bottom: 0.3em; font-weight: bold; }
dl dd { margin-bottom: 0.75em; }

/* Abbreviations */
abbr, acronym { text-transform: uppercase; font-size: 90%; color: black; border-bottom: 1px dotted #dddddd; cursor: help; }

abbr { text-transform: none; }

/* Blockquotes */
blockquote { margin: 0 0 0.75em; padding: 0.5625em 1.25em 0 1.1875em; border-left: 1px solid #dddddd; }
blockquote cite { display: block; font-size: 0.8125em; color: #5e93b8; }
blockquote cite:before { content: "\2014 \0020"; }
blockquote cite a, blockquote cite a:visited { color: #5e93b8; }

blockquote, blockquote p { line-height: 1.6; color: #333333; }

/* Microformats */
.vcard { display: inline-block; margin: 0 0 1.25em 0; border: 1px solid #dddddd; padding: 0.625em 0.75em; }
.vcard li { margin: 0; display: block; }
.vcard .fn { font-weight: bold; font-size: 0.9375em; }

.vevent .summary { font-weight: bold; }
.vevent abbr { cursor: auto; text-decoration: none; font-weight: bold; border: none; padding: 0 0.0625em; }

@media only screen and (min-width: 768px) { h1, h2, h3, #toctitle, .sidebarblock > .content > .title, h4, h5, h6 { line-height: 1.4; }
  h1 { font-size: 2.75em; }
  h2 { font-size: 2.3125em; }
  h3, #toctitle, .sidebarblock > .content > .title { font-size: 1.6875em; }
  h4 { font-size: 1.4375em; } }
/* Tables */
table { background: white; margin-bottom: 1.25em; border: solid 1px #d8d8ce; }
table thead, table tfoot { background: -webkit-linear-gradient(top, #add386, #90b66a); font-weight: bold; }
table thead tr th, table thead tr td, table tfoot tr th, table tfoot tr td { padding: 0.5em 0.625em 0.625em; font-size: inherit; color: white; text-align: left; }
table tr th, table tr td { padding: 0.5625em 0.625em; font-size: inherit; color: #6d6e71; }
table tr.even, table tr.alt, table tr:nth-of-type(even) { background: #edf2f2; }
table thead tr th, table tfoot tr th, table tbody tr td, table tr td, table tfoot tr td { display: table-cell; line-height: 1.4; }

body { -moz-osx-font-smoothing: grayscale; -webkit-font-smoothing: antialiased; tab-size: 4; }

h1, h2, h3, #toctitle, .sidebarblock > .content > .title, h4, h5, h6 { line-height: 1.4; }

a:hover, a:focus { text-decoration: underline; }

.clearfix:before, .clearfix:after, .float-group:before, .float-group:after { content: " "; display: table; }
.clearfix:after, .float-group:after { clear: both; }

*:not(pre) > code { font-size: inherit; font-style: normal !important; letter-spacing: 0; padding: 0; background-color: white; -webkit-border-radius: 0; border-radius: 0; line-height: inherit; word-wrap: break-word; }
*:not(pre) > code.nobreak { word-wrap: normal; }
*:not(pre) > code.nowrap { white-space: nowrap; }

pre, pre > code { line-height: 1.6; color: #264357; font-family: Consolas, "Liberation Mono", Courier, monospace; font-weight: normal; }

em em { font-style: normal; }

strong strong { font-weight: normal; }

.keyseq { color: #333333; }

kbd { font-family: Consolas, "Liberation Mono", Courier, monospace; display: inline-block; color: black; font-size: 0.65em; line-height: 1.45; background-color: #f7f7f7; border: 1px solid #ccc; -webkit-border-radius: 3px; border-radius: 3px; -webkit-box-shadow: 0 1px 0 rgba(0, 0, 0, 0.2), 0 0 0 0.1em white inset; box-shadow: 0 1px 0 rgba(0, 0, 0, 0.2), 0 0 0 0.1em white inset; margin: 0 0.15em; padding: 0.2em 0.5em; vertical-align: middle; position: relative; top: -0.1em; white-space: nowrap; }

.keyseq kbd:first-child { margin-left: 0; }

.keyseq kbd:last-child { margin-right: 0; }

.menuseq, .menuref { color: #000; }

.menuseq b:not(.caret), .menuref { font-weight: inherit; }

.menuseq { word-spacing: -0.02em; }
.menuseq b.caret { font-size: 1.25em; line-height: 0.8; }
.menuseq i.caret { font-weight: bold; text-align: center; width: 0.45em; }

b.button:before, b.button:after { position: relative; top: -1px; font-weight: normal; }

b.button:before { content: "["; padding: 0 3px 0 2px; }

b.button:after { content: "]"; padding: 0 2px 0 3px; }

#header, #content, #footnotes, #footer { width: 100%; margin-left: auto; margin-right: auto; margin-top: 0; margin-bottom: 0; max-width: 62.5em; *zoom: 1; position: relative; padding-left: 1.5em; padding-right: 1.5em; }
#header:before, #header:after, #content:before, #content:after, #footnotes:before, #footnotes:after, #footer:before, #footer:after { content: " "; display: table; }
#header:after, #content:after, #footnotes:after, #footer:after { clear: both; }

#content { margin-top: 1.25em; }

#content:before { content: none; }

#header > h1:first-child { color: black; margin-top: 2.25rem; margin-bottom: 0; }
#header > h1:first-child + #toc { margin-top: 8px; border-top: 1px solid #dddddd; }
#header > h1:only-child, body.toc2 #header > h1:nth-last-child(2) { border-bottom: 1px solid #dddddd; padding-bottom: 8px; }
#header .details { border-bottom: 1px solid #dddddd; line-height: 1.45; padding-top: 0.25em; padding-bottom: 0.25em; padding-left: 0.25em; color: #5e93b8; display: -ms-flexbox; display: -webkit-flex; display: flex; -ms-flex-flow: row wrap; -webkit-flex-flow: row wrap; flex-flow: row wrap; }
#header .details span:first-child { margin-left: -0.125em; }
#header .details span.email a { color: #333333; }
#header .details br { display: none; }
#header .details br + span:before { content: "\00a0\2013\00a0"; }
#header .details br + span.author:before { content: "\00a0\22c5\00a0"; color: #333333; }
#header .details br + span#revremark:before { content: "\00a0|\00a0"; }
#header #revnumber { text-transform: capitalize; }
#header #revnumber:after { content: "\00a0"; }

#content > h1:first-child:not([class]) { color: black; border-bottom: 1px solid #dddddd; padding-bottom: 8px; margin-top: 0; padding-top: 1rem; margin-bottom: 1.25rem; }

#toc { border-bottom: 0 solid #dddddd; padding-bottom: 0.5em; }
#toc > ul { margin-left: 0.125em; }
#toc ul.sectlevel0 > li > a { font-style: italic; }
#toc ul.sectlevel0 ul.sectlevel1 { margin: 0.5em 0; }
#toc ul { font-family: Noto, sans-serif; list-style-type: none; }
#toc li { line-height: 1.3334; margin-top: 0.3334em; }
#toc a { text-decoration: none; }
#toc a:active { text-decoration: underline; }

#toctitle { color: black; font-size: 1.2em; }

@media only screen and (min-width: 768px) { #toctitle { font-size: 1.375em; }
  body.toc2 { padding-left: 15em; padding-right: 0; }
  #toc.toc2 { margin-top: 0 !important; background-color: white; position: fixed; width: 15em; left: 0; top: 0; border-right: 1px solid #dddddd; border-top-width: 0 !important; border-bottom-width: 0 !important; z-index: 1000; padding: 1.25em 1em; height: 100%; overflow: auto; }
  #toc.toc2 #toctitle { margin-top: 0; margin-bottom: 0.8rem; font-size: 1.2em; }
  #toc.toc2 > ul { font-size: 0.9em; margin-bottom: 0; }
  #toc.toc2 ul ul { margin-left: 0; padding-left: 1em; }
  #toc.toc2 ul.sectlevel0 ul.sectlevel1 { padding-left: 0; margin-top: 0.5em; margin-bottom: 0.5em; }
  body.toc2.toc-right { padding-left: 0; padding-right: 15em; }
  body.toc2.toc-right #toc.toc2 { border-right-width: 0; border-left: 1px solid #dddddd; left: auto; right: 0; } }
@media only screen and (min-width: 1280px) { body.toc2 { padding-left: 20em; padding-right: 0; }
  #toc.toc2 { width: 20em; }
  #toc.toc2 #toctitle { font-size: 1.375em; }
  #toc.toc2 > ul { font-size: 0.95em; }
  #toc.toc2 ul ul { padding-left: 1.25em; }
  body.toc2.toc-right { padding-left: 0; padding-right: 20em; } }
#content #toc { border-style: solid; border-width: 1px; border-color: #e6e6e6; margin-bottom: 1.25em; padding: 1.25em; background: white; -webkit-border-radius: 0; border-radius: 0; }
#content #toc > :first-child { margin-top: 0; }
#content #toc > :last-child { margin-bottom: 0; }

#footer { max-width: 100%; background-color: none; padding: 1.25em; }

#footer-text { color: black; line-height: 1.44; }

#content { margin-bottom: 0.625em; }

.sect1 { padding-bottom: 0.625em; }

@media only screen and (min-width: 768px) { #content { margin-bottom: 1.25em; }
  .sect1 { padding-bottom: 1.25em; } }
.sect1:last-child { padding-bottom: 0; }

.sect1 + .sect1 { border-top: 0 solid #dddddd; }

#content h1 > a.anchor, h2 > a.anchor, h3 > a.anchor, #toctitle > a.anchor, .sidebarblock > .content > .title > a.anchor, h4 > a.anchor, h5 > a.anchor, h6 > a.anchor { position: absolute; z-index: 1001; width: 1.5ex; margin-left: -1.5ex; display: block; text-decoration: none !important; visibility: hidden; text-align: center; font-weight: normal; }
#content h1 > a.anchor:before, h2 > a.anchor:before, h3 > a.anchor:before, #toctitle > a.anchor:before, .sidebarblock > .content > .title > a.anchor:before, h4 > a.anchor:before, h5 > a.anchor:before, h6 > a.anchor:before { content: "\00A7"; font-size: 0.85em; display: block; padding-top: 0.1em; }
#content h1:hover > a.anchor, #content h1 > a.anchor:hover, h2:hover > a.anchor, h2 > a.anchor:hover, h3:hover > a.anchor, #toctitle:hover > a.anchor, .sidebarblock > .content > .title:hover > a.anchor, h3 > a.anchor:hover, #toctitle > a.anchor:hover, .sidebarblock > .content > .title > a.anchor:hover, h4:hover > a.anchor, h4 > a.anchor:hover, h5:hover > a.anchor, h5 > a.anchor:hover, h6:hover > a.anchor, h6 > a.anchor:hover { visibility: visible; }
#content h1 > a.link, h2 > a.link, h3 > a.link, #toctitle > a.link, .sidebarblock > .content > .title > a.link, h4 > a.link, h5 > a.link, h6 > a.link { color: black; text-decoration: none; }
#content h1 > a.link:hover, h2 > a.link:hover, h3 > a.link:hover, #toctitle > a.link:hover, .sidebarblock > .content > .title > a.link:hover, h4 > a.link:hover, h5 > a.link:hover, h6 > a.link:hover { color: black; }

.audioblock, .imageblock, .literalblock, .listingblock, .stemblock, .videoblock { margin-bottom: 1.25em; }

.admonitionblock td.content > .title, .audioblock > .title, .exampleblock > .title, .imageblock > .title, .listingblock > .title, .literalblock > .title, .stemblock > .title, .openblock > .title, .paragraph > .title, .quoteblock > .title, table.tableblock > .title, .verseblock > .title, .videoblock > .title, .dlist > .title, .olist > .title, .ulist > .title, .qlist > .title, .hdlist > .title { text-rendering: optimizeLegibility; text-align: left; }

table.tableblock > caption.title { white-space: nowrap; overflow: visible; max-width: 0; }

.paragraph.lead > p, #preamble > .sectionbody > .paragraph:first-of-type p { color: black; }

table.tableblock #preamble > .sectionbody > .paragraph:first-of-type p { font-size: inherit; }

.admonitionblock > table { border-collapse: separate; border: 0; background: none; width: 100%; }
.admonitionblock > table td.icon { text-align: center; width: 80px; }
.admonitionblock > table td.icon img { max-width: initial; }
.admonitionblock > table td.icon .title { font-weight: bold; font-family: Noto, sans-serif; text-transform: uppercase; }
.admonitionblock > table td.content { padding-left: 1.125em; padding-right: 1.25em; border-left: 1px solid #dddddd; color: #5e93b8; }
.admonitionblock > table td.content > :last-child > :last-child { margin-bottom: 0; }

.exampleblock > .content { border-style: solid; border-width: 1px; border-color: #e6e6e6; margin-bottom: 1.25em; padding: 1.25em; background: white; -webkit-border-radius: 0; border-radius: 0; }
.exampleblock > .content > :first-child { margin-top: 0; }
.exampleblock > .content > :last-child { margin-bottom: 0; }

.sidebarblock { border-style: solid; border-width: 1px; border-color: #e6e6e6; margin-bottom: 1.25em; padding: 1.25em; background: white; -webkit-border-radius: 0; border-radius: 0; }
.sidebarblock > :first-child { margin-top: 0; }
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<body class="manpage">
<div id="header">
<h1>mathFunctions(3) Manual Page</h1>
<h2 id="_name">Name</h2>
<div class="sectionbody">
<p>mathFunctions - Math Functions</p>
</div>
</div>
<div id="content">
<div class="sect1">
<h2 id="_description">Description</h2>
<div class="sectionbody">
<div class="paragraph">
<p>The built-in math functions are categorized into the following:</p>
</div>
<div class="ulist">
<ul>
<li>
<p>A list of built-in functions that have scalar or vector argument
versions, and,</p>
</li>
<li>
<p>A list of built-in functions that only take scalar <code>float</code> arguments.</p>
</li>
</ul>
</div>
<div class="paragraph">
<p>The vector versions of the math functions operate component-wise.
The description is per-component.</p>
</div>
<div class="paragraph">
<p>The built-in math functions are not affected by the prevailing rounding mode
in the calling environment, and always return the same value as they would
if called with the round to nearest even rounding mode.</p>
</div>
<div class="paragraph">
<p>The <a href="https://www.khronos.org/registry/OpenCL/specs/2.2/html/OpenCL_C.html#table-builtin-math" target="_blank" rel="noopener">following table</a> describes the list of built-in
math functions that can take scalar or vector arguments.
We use the generic type name <code>gentype</code> to indicate that the function can
take <code>float</code>, <code>float2</code>, <code>float3</code>, <code>float4</code>, <code>float8</code>, <code>float16</code>, <code>double</code>,
<code>double2</code>, <code>double3</code>, <code>double4</code>, <code>double8</code> or <code>double16</code> as the type for the
arguments.
We use the generic type name <code>gentypef</code> to indicate that the function can
take <code>float</code>, <code>float2</code>, <code>float3</code>, <code>float4</code>, <code>float8</code>, or <code>float16</code> as the
type for the arguments.
We use the generic type name <code>gentyped</code> to indicate that the function can
take <code>double</code>, <code>double2</code>, <code>double3</code>, <code>double4</code>, <code>double8</code> or <code>double16</code> as
the type for the arguments.
For any specific use of a function, the actual type has to be the same for
all arguments and the return type, unless otherwise specified.</p>
</div>
<table id="table-builtin-math" class="tableblock frame-all grid-all stretch">
<caption class="title">Table 1. Scalar and Vector Argument Built-in Math Function Table</caption>
<colgroup>
<col style="width: 50%;">
<col style="width: 50%;">
</colgroup>
<tbody>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>Function</strong></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>Description</strong></p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>acos</strong>(gentype)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Arc cosine function. Returns an angle in radians.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>acosh</strong>(gentype)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Inverse hyperbolic cosine. Returns an angle in radians.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>acospi</strong>(gentype <em>x</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute <strong>acos</strong>(<em>x</em>) / π.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>asin</strong>(gentype)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Arc sine function. Returns an angle in radians.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>asinh</strong>(gentype)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Inverse hyperbolic sine. Returns an angle in radians.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>asinpi</strong>(gentype <em>x</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute <strong>asin</strong>(<em>x</em>) / π.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>atan</strong>(gentype <em>y_over_x</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Arc tangent function. Returns an angle in radians.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>atan2</strong>(gentype <em>y</em>, gentype <em>x</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Arc tangent of <em>y</em> / <em>x</em>. Returns an angle in radians.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>atanh</strong>(gentype)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Hyperbolic arc tangent. Returns an angle in radians.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>atanpi</strong>(gentype <em>x</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute <strong>atan</strong>(<em>x</em>) / π.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>atan2pi</strong>(gentype <em>y</em>, gentype <em>x</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute <strong>atan2</strong>(<em>y</em>, <em>x</em>) / π.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>cbrt</strong>(gentype)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute cube-root.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>ceil</strong>(gentype)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Round to integral value using the round to positive infinity rounding
      mode.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>copysign</strong>(gentype <em>x</em>, gentype <em>y</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Returns <em>x</em> with its sign changed to match the sign of <em>y</em>.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>cos</strong>(gentype <em>x</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute cosine, where <em>x</em> is an angle in radians.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>cosh</strong>(gentype <em>x</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute hyperbolic cosine, where <em>x</em> is an angle in radians.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>cospi</strong>(gentype <em>x</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute <strong>cos</strong>(π <em>x</em>).</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>erfc</strong>(gentype)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Complementary error function.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>erf</strong>(gentype)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Error function encountered in integrating the
      <a href="http://mathworld.wolfram.com/NormalDistribution.html"><em>normal
      distribution</em></a>.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>exp</strong>(gentype <em>x</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute the base-<em>e</em> exponential of <em>x</em>.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>exp2</strong>(gentype)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Exponential base 2 function.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>exp10</strong>(gentype)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Exponential base 10 function.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>expm1</strong>(gentype <em>x</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute <em>e<sup>x</sup></em> - 1.0.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>fabs</strong>(gentype)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute absolute value of a floating-point number.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>fdim</strong>(gentype <em>x</em>, gentype <em>y</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock"><em>x</em> - <em>y</em> if <em>x</em> &gt; <em>y</em>, +0 if <em>x</em> is less than or equal to y.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>floor</strong>(gentype)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Round to integral value using the round to negative infinity rounding
      mode.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>fma</strong>(gentype <em>a</em>, gentype <em>b</em>, gentype <em>c</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the correctly rounded floating-point representation of the sum
      of <em>c</em> with the infinitely precise product of <em>a</em> and <em>b</em>.
      Rounding of intermediate products shall not occur.
      Edge case behavior is per the IEEE 754-2008 standard.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>fmax</strong>(gentype <em>x</em>, gentype <em>y</em>)<br>
  gentypef <strong>fmax</strong>(gentypef <em>x</em>, float <em>y</em>)<br>
  gentyped <strong>fmax</strong>(gentyped <em>x</em>, double <em>y</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Returns <em>y</em> if <em>x</em> &lt; <em>y</em>, otherwise it returns <em>x</em>.
      If one argument is a NaN, <strong>fmax</strong>() returns the other argument.
      If both arguments are NaNs, <strong>fmax</strong>() returns a NaN.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>fmin</strong><sup>29</sup>(gentype <em>x</em>, gentype <em>y</em>)<br>
  gentypef <strong>fmin</strong>(gentypef <em>x</em>, float <em>y</em>)<br>
  gentyped <strong>fmin</strong>(gentyped <em>x</em>, double <em>y</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Returns <em>y</em> if <em>y</em> &lt; <em>x</em>, otherwise it returns <em>x</em>.
      If one argument is a NaN, <strong>fmin</strong>() returns the other argument.
      If both arguments are NaNs, <strong>fmin</strong>() returns a NaN.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>fmod</strong>(gentype <em>x</em>, gentype <em>y</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Modulus.
      Returns <em>x</em> - <em>y</em> * <strong>trunc</strong>(<em>x</em>/<em>y</em>).</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>fract</strong>(gentype <em>x</em>, gentype <em>*iptr</em>)<sup>30</sup></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Returns <strong>fmin</strong>(<em>x</em> - <strong>floor</strong>(<em>x</em>), <code>0x1.fffffep-1f</code>).
      <strong>floor</strong>(x) is returned in <em>iptr</em>.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">float<em>n</em> <strong>frexp</strong>(float<em>n</em> <em>x</em>, int<em>n</em> *exp)<br>
  float <strong>frexp</strong>(float <em>x</em>, int *exp)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Extract mantissa and exponent from <em>x</em>.
      For each component the mantissa returned is a <code>float</code> with magnitude
      in the interval [1/2, 1) or 0.
      Each component of <em>x</em> equals mantissa returned * 2<em><sup>exp</sup></em>.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">double<em>n</em> <strong>frexp</strong>(double<em>n</em> <em>x</em>, int<em>n</em> *exp)<br>
  double <strong>frexp</strong>(double <em>x</em>, int *exp)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Extract mantissa and exponent from <em>x</em>.
      For each component the mantissa returned is a <code>float</code> with magnitude
      in the interval [1/2, 1) or 0.
      Each component of <em>x</em> equals mantissa returned * 2<em><sup>exp</sup></em>.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>hypot</strong>(gentype <em>x</em>, gentype <em>y</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute the value of the square root of <em>x</em><sup>2</sup>+ <em>y</em><sup>2</sup> without
      undue overflow or underflow.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">int<em>n</em> <strong>ilogb</strong>(float<em>n</em> <em>x</em>)<br>
  int <strong>ilogb</strong>(float <em>x</em>)<br>
  int<em>n</em> <strong>ilogb</strong>(double<em>n</em> <em>x</em>)<br>
  int <strong>ilogb</strong>(double <em>x</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Return the exponent as an integer value.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">float<em>n</em> <strong>ldexp</strong>(float<em>n</em> <em>x</em>, int<em>n</em> <em>k</em>)<br>
  float<em>n</em> <strong>ldexp</strong>(float<em>n</em> <em>x</em>, int <em>k</em>)<br>
  float <strong>ldexp</strong>(float <em>x</em>, int <em>k</em>)<br>
  double<em>n</em> <strong>ldexp</strong>(double<em>n</em> <em>x</em>, int<em>n</em> <em>k</em>)<br>
  double<em>n</em> <strong>ldexp</strong>(double<em>n</em> <em>x</em>, int <em>k</em>)<br>
  double <strong>ldexp</strong>(double <em>x</em>, int <em>k</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Multiply <em>x</em> by 2 to the power <em>k</em>.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>lgamma</strong>(gentype <em>x</em>)<br>
  float<em>n</em> <strong>lgamma_r</strong>(float<em>n</em> <em>x</em>, int<em>n</em> *<em>signp</em>)<br>
  float <strong>lgamma_r</strong>(float <em>x</em>, int *<em>signp</em>)<br>
  double<em>n</em> <strong>lgamma_r</strong>(double<em>n</em> <em>x</em>, int<em>n</em> *<em>signp</em>)<br>
  double <strong>lgamma_r</strong>(double <em>x</em>, int *<em>signp</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Log gamma function.
      Returns the natural logarithm of the absolute value of the gamma
      function.
      The sign of the gamma function is returned in the <em>signp</em> argument of
      <strong>lgamma_r</strong>.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>log</strong>(gentype)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute natural logarithm.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>log2</strong>(gentype)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute a base 2 logarithm.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>log10</strong>(gentype)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute a base 10 logarithm.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>log1p</strong>(gentype <em>x</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute log<sub>e</sub>(1.0 + <em>x</em>).</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>logb</strong>(gentype <em>x</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute the exponent of <em>x</em>, which is the integral part of
      log<em><sub>r</sub></em>(|<em>x</em>|).</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>mad</strong>(gentype <em>a</em>, gentype <em>b</em>, gentype <em>c</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>mad</strong> computes <em>a</em> * <em>b</em> + <em>c</em>.
    The function may compute <em>a</em> * <em>b</em> + <em>c</em> with reduced accuracy
    in the embedded profile.  See the SPIR-V OpenCL environment specification
    for details. On some hardware the mad instruction may provide better
    performance than expanded computation of <em>a</em> * <em>b</em> + <em>c</em>.<sup>31</sup></p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>maxmag</strong>(gentype <em>x</em>, gentype <em>y</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Returns <em>x</em> if |<em>x</em>| &gt; |<em>y</em>|, <em>y</em> if |<em>y</em>| &gt; |<em>x</em>|, otherwise
      <strong>fmax</strong>(<em>x</em>, <em>y</em>).</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>minmag</strong>(gentype <em>x</em>, gentype <em>y</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Returns <em>x</em> if |<em>x</em>| &lt; |<em>y</em>|, <em>y</em> if |<em>y</em>| &lt; |<em>x</em>|, otherwise
      <strong>fmin</strong>(<em>x</em>, <em>y</em>).</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>modf</strong>(gentype <em>x</em>, gentype *<em>iptr</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Decompose a floating-point number.
      The <strong>modf</strong> function breaks the argument <em>x</em> into integral and
      fractional parts, each of which has the same sign as the argument.
      It stores the integral part in the object pointed to by <em>iptr</em>.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">float<em>n</em> <strong>nan</strong>(uint<em>n</em> <em>nancode</em>)<br>
  float <strong>nan</strong>(uint <em>nancode</em>)<br>
  double<em>n</em> <strong>nan</strong>(ulong<em>n</em> <em>nancode</em>)<br>
  double <strong>nan</strong>(ulong <em>nancode</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Returns a quiet NaN.
      The <em>nancode</em> may be placed in the significand of the resulting NaN.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>nextafter</strong>(gentype <em>x</em>, gentype <em>y</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Computes the next representable single-precision floating-point value
      following <em>x</em> in the direction of <em>y</em>.
      Thus, if <em>y</em> is less than <em>x</em>, <strong>nextafter</strong>() returns the largest
      representable floating-point number less than <em>x</em>.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>pow</strong>(gentype <em>x</em>, gentype <em>y</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute <em>x</em> to the power <em>y</em>.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">float<em>n</em> <strong>pown</strong>(float<em>n</em> <em>x</em>, int<em>n</em> <em>y</em>)<br>
  float <strong>pown</strong>(float <em>x</em>, int <em>y</em>)<br>
  double<em>n</em> <strong>pown</strong>(double<em>n</em> <em>x</em>, int<em>n</em> <em>y</em>)<br>
  double <strong>pown</strong>(double <em>x</em>, int <em>y</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute <em>x</em> to the power <em>y</em>, where <em>y</em> is an integer.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>powr</strong>(gentype <em>x</em>, gentype <em>y</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute <em>x</em> to the power <em>y</em>, where <em>x</em> is &gt;= 0.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>remainder</strong>(gentype <em>x</em>, gentype <em>y</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute the value <em>r</em> such that <em>r</em> = <em>x</em> - <em>n</em>*<em>y</em>, where <em>n</em> is the
      integer nearest the exact value of <em>x</em>/<em>y</em>.
      If there are two integers closest to <em>x</em>/<em>y</em>, <em>n</em> shall be the even
      one.
      If <em>r</em> is zero, it is given the same sign as <em>x</em>.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">float<em>n</em> <strong>remquo</strong>(float<em>n</em> <em>x</em>, float<em>n</em> <em>y</em>, int<em>n</em> *<em>quo</em>)<br>
  float <strong>remquo</strong>(float <em>x</em>, float <em>y</em>, int *<em>quo</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">The <strong>remquo</strong> function computes the value r such that <em>r</em> = <em>x</em> -
      <em>k</em>*<em>y</em>, where <em>k</em> is the integer nearest the exact value of <em>x</em>/<em>y</em>.
      If there are two integers closest to <em>x</em>/<em>y</em>, <em>k</em> shall be the even
      one.
      If <em>r</em> is zero, it is given the same sign as <em>x</em>.
      This is the same value that is returned by the <strong>remainder</strong> function.
      <strong>remquo</strong> also calculates the lower seven bits of the integral quotient
      <em>x</em>/<em>y</em>, and gives that value the same sign as <em>x</em>/<em>y</em>.
      It stores this signed value in the object pointed to by <em>quo</em>.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">double<em>n</em> <strong>remquo</strong>(double<em>n</em> <em>x</em>, double<em>n</em> <em>y</em>, int<em>n</em> *<em>quo</em>)<br>
  double <strong>remquo</strong>(double <em>x</em>, double <em>y</em>, int *<em>quo</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">The <strong>remquo</strong> function computes the value r such that <em>r</em> = <em>x</em> -
      <em>k</em>*<em>y</em>, where <em>k</em> is the integer nearest the exact value of <em>x</em>/<em>y</em>.
      If there are two integers closest to <em>x</em>/<em>y</em>, <em>k</em> shall be the even
      one.
      If <em>r</em> is zero, it is given the same sign as <em>x</em>.
      This is the same value that is returned by the <strong>remainder</strong> function.
      <strong>remquo</strong> also calculates the lower seven bits of the integral quotient
      <em>x</em>/<em>y</em>, and gives that value the same sign as <em>x</em>/<em>y</em>.
      It stores this signed value in the object pointed to by <em>quo</em>.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>rint</strong>(gentype)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Round to integral value (using round to nearest even rounding mode) in
      floating-point format.
      Refer to section 7.1 for description of rounding modes.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">float<em>n</em> <strong>rootn</strong>(float<em>n</em> <em>x</em>, int<em>n</em> <em>y</em>)<br>
  float <strong>rootn</strong>(float <em>x</em>, int <em>y</em>)<br>
  double<em>n</em> <strong>rootn</strong>(double<em>n</em> <em>x</em>, int<em>n</em> <em>y</em>)<br>
  double<em>n</em> <strong>rootn</strong>(double <em>x</em>, int <em>y</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute <em>x</em> to the power 1/<em>y</em>.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>round</strong>(gentype <em>x</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Return the integral value nearest to <em>x</em> rounding halfway cases away
      from zero, regardless of the current rounding direction.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>rsqrt</strong>(gentype)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute inverse square root.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>sin</strong>(gentype <em>x</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute sine, where <em>x</em> is an angle in radians.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>sincos</strong>(gentype <em>x</em>, gentype *<em>cosval</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute sine and cosine of x.
      The computed sine is the return value and computed cosine is returned
      in <em>cosval</em>, where <em>x</em> is an angle in radians.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>sinh</strong>(gentype <em>x</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute hyperbolic sine, where <em>x</em> is an angle in radians</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>sinpi</strong>(gentype <em>x</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute <strong>sin</strong>(π <em>x</em>).</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>sqrt</strong>(gentype)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute square root.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>tan</strong>(gentype <em>x</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute tangent, where <em>x</em> is an angle in radians.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>tanh</strong>(gentype <em>x</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute hyperbolic tangent, where <em>x</em> is an angle in radians.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>tanpi</strong>(gentype <em>x</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute <strong>tan</strong>(π <em>x</em>).</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>tgamma</strong>(gentype)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute the gamma function.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>trunc</strong>(gentype)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Round to integral value using the round to zero rounding mode.</p></td>
</tr>
</tbody>
</table>
<div class="paragraph">
<p>[29] <strong>fmin</strong> and <strong>fmax</strong> behave as defined by C99 and may not match the IEEE
754-2008 definition for <strong>minNum</strong> and <strong>maxNum</strong> with regard to signaling NaNs.
Specifically, signaling NaNs may behave as quiet NaNs.</p>
</div>
<div class="paragraph">
<p>[30] The <strong>min</strong>() operator is there to prevent <strong>fract</strong>(-small) from returning
1.0.
It returns the largest positive floating-point number less than 1.0.</p>
</div>
<div class="paragraph">
<p>[31] The user is cautioned that for some usages, e.g. <strong>mad</strong>(a, b, -a*b), the
definition of <strong>mad</strong>() is loose enough in the embedded profile that almost any result is allowed from
<strong>mad</strong>() for some values of a and b.</p>
</div>
<div class="paragraph">
<p>The following table describes the following functions:</p>
</div>
<div class="ulist">
<ul>
<li>
<p>A subset of functions from <a href="https://www.khronos.org/registry/OpenCL/specs/2.2/html/OpenCL_C.html#table-builtin-math" class="bare" target="_blank" rel="noopener">https://www.khronos.org/registry/OpenCL/specs/2.2/html/OpenCL_C.html#table-builtin-math</a> that are defined with
the half_ prefix .
These functions are implemented with a minimum of 10-bits of accuracy,
i.e. an ULP value &lt;= 8192 ulp.</p>
</li>
<li>
<p>A subset of functions from <a href="https://www.khronos.org/registry/OpenCL/specs/2.2/html/OpenCL_C.html#table-builtin-math" class="bare" target="_blank" rel="noopener">https://www.khronos.org/registry/OpenCL/specs/2.2/html/OpenCL_C.html#table-builtin-math</a> that are defined with
the native_ prefix.
These functions may map to one or more native device instructions and
will typically have better performance compared to the corresponding
functions (without the <code>native_</code> prefix) described in
<a href="https://www.khronos.org/registry/OpenCL/specs/2.2/html/OpenCL_C.html#table-builtin-math" class="bare" target="_blank" rel="noopener">https://www.khronos.org/registry/OpenCL/specs/2.2/html/OpenCL_C.html#table-builtin-math</a>.
The accuracy (and in some cases the input range(s)) of these functions
is implementation-defined.</p>
</li>
<li>
<p><code>half_</code> and <code>native_</code> functions for following basic operations:
divide and reciprocal.</p>
</li>
</ul>
</div>
<div class="paragraph">
<p>We use the generic type name <code>gentype</code> to indicate that the functions in the
following table can take <code>float</code>, <code>float2</code>, <code>float3</code>, <code>float4</code>, <code>float8</code> or
<code>float16</code> as the type for the arguments.</p>
</div>
<table id="table-builtin-half-native-math" class="tableblock frame-all grid-all stretch">
<caption class="title">Table 2. Scalar and Vector Argument Built-in <em>half</em> and <em>native</em> Math Functions</caption>
<colgroup>
<col style="width: 50%;">
<col style="width: 50%;">
</colgroup>
<tbody>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>Function</strong></p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>Description</strong></p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>half_cos</strong>(gentype <em>x</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute cosine.
      <em>x</em> is an angle in radians, and must be in the range [-2<sup>16</sup>, +2<sup>16</sup>].</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>half_divide</strong>(gentype <em>x</em>, gentype <em>y</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute <em>x</em> / <em>y</em>.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>half_exp</strong>(gentype <em>x</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute the base-<em>e</em> exponential of <em>x</em>.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>half_exp2</strong>(gentype <em>x</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute the base- 2 exponential of <em>x</em>.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>half_exp10</strong>(gentype <em>x</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute the base- 10 exponential of <em>x</em>.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>half_log</strong>(gentype <em>x</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute natural logarithm.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>half_log2</strong>(gentype <em>x</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute a base 2 logarithm.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>half_log10</strong>(gentype <em>x</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute a base 10 logarithm.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>half_powr</strong>(gentype <em>x</em>, gentype <em>y</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute <em>x</em> to the power <em>y</em>, where <em>x</em> is &gt;= 0.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>half_recip</strong>(gentype <em>x</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute reciprocal.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>half_rsqrt</strong>(gentype <em>x</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute inverse square root.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>half_sin</strong>(gentype <em>x</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute sine.
      <em>x</em> is an angle in radians, and must be in the range [-2<sup>16</sup>, +2<sup>16</sup>].</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>half_sqrt</strong>(gentype <em>x</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute square root.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>half_tan</strong>(gentype <em>x</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute tangent.
      <em>x</em> is an angle in radians, and must be in the range [-2<sup>16</sup>, +2<sup>16</sup>].</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"></td>
<td class="tableblock halign-left valign-top"></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>native_cos</strong>(gentype <em>x</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute cosine over an implementation-defined range, where <em>x</em> is an
      angle in radians.
      The maximum error is implementation-defined.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>native_divide</strong>(gentype <em>x</em>, gentype <em>y</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute <em>x</em> / <em>y</em> over an implementation-defined range.
      The maximum error is implementation-defined.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>native_exp</strong>(gentype <em>x</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute the base-<em>e</em> exponential of <em>x</em> over an
      implementation-defined range.
      The maximum error is implementation-defined.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>native_exp2</strong>(gentype <em>x</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute the base-2 exponential of <em>x</em> over an implementation-defined
      range.
      The maximum error is implementation-defined.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>native_exp10</strong>(gentype <em>x</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute the base-10 exponential of <em>x</em> over an implementation-defined
      range.
      The maximum error is implementation-defined.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>native_log</strong>(gentype <em>x</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute natural logarithm over an implementation-defined range.
      The maximum error is implementation-defined.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>native_log2</strong>(gentype <em>x</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute a base 2 logarithm over an
      implementation-defined range.
      The maximum error is implementation-defined.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>native_log10</strong>(gentype <em>x</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute a base 10 logarithm over
      an implementation-defined range.
      The maximum error is implementation-defined.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>native_powr</strong>(gentype <em>x</em>, gentype <em>y</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute <em>x</em> to the power <em>y</em>, where <em>x</em> is &gt;= 0.
      The range of <em>x</em> and <em>y</em> are implementation-defined.
      The maximum error is implementation-defined.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>native_recip</strong>(gentype <em>x</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute reciprocal over an implementation-defined range.
      The maximum error is implementation-defined.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>native_rsqrt</strong>(gentype <em>x</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute inverse square root over an implementation-defined range.
      The maximum error is implementation-defined.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>native_sin</strong>(gentype <em>x</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute sine over an implementation-defined range, where <em>x</em> is an
      angle in radians.
      The maximum error is implementation-defined.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>native_sqrt</strong>(gentype <em>x</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute square root over an implementation-defined range.
      The maximum error is implementation-defined.</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">gentype <strong>native_tan</strong>(gentype <em>x</em>)</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">Compute tangent over an implementation-defined range, where <em>x</em> is an
      angle in radians.
      The maximum error is implementation-defined.</p></td>
</tr>
</tbody>
</table>
<div class="paragraph">
<p>Support for denormal values is optional for <strong>half_</strong> functions.
The <strong>half_</strong> functions may return any result allowed by
<a href="https://www.khronos.org/registry/OpenCL/specs/2.2/html/OpenCL_C.html#edge-case-behavior-in-flush-to-zero-mode" target="_blank" rel="noopener">Edge Case Behavior</a>, even when
<code>-cl-denorms-are-zero</code> (see <a href="https://www.khronos.org/registry/OpenCL/specs/2.2/html/OpenCL_C.html#opencl-spec" target="_blank" rel="noopener">section 5.8.4.2 of the OpenCL
Specification</a>) is not in force.
Support for denormal values is implementation-defined for <strong>native_</strong>
functions.</p>
</div>
</div>
</div>
<div class="sect1">
<h2 id="_see_also">See Also</h2>
<div class="sectionbody">
<div class="paragraph">
<p><a href="commonFunctions.html">commonFunctions</a>, <a href="integerFunctions.html">integerFunctions</a></p>
</div>
</div>
</div>
<div class="sect1">
<h2 id="_document_notes">Document Notes</h2>
<div class="sectionbody">
<div class="paragraph">
<p>For more information, see the <a href="https://www.khronos.org/registry/OpenCL/specs/2.2/html/OpenCL_C.html#math-functions" target="_blank" rel="noopener">OpenCL C Specification</a></p>
</div>
<div class="paragraph">
<p>This page is extracted from the OpenCL C Specification.
Fixes and changes should be made to the Specification, not directly.</p>
</div>
</div>
</div>
<div class="sect1">
<h2 id="_copyright">Copyright</h2>
<div class="sectionbody">
<div class="paragraph">
<p>Copyright (c) 2014-2020 Khronos Group. This work is licensed under a
<a href="http://creativecommons.org/licenses/by/4.0/" target="_blank" rel="noopener">Creative Commons
Attribution 4.0 International License</a>.</p>
</div>
</div>
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